A common approach to analysing the final size of an SIR epidemic is via a suitable random time-scale transformation which leaves the final size unchanged.
For the present model this yields a deterministic limiting process that is not Lipschitz in the vicinity of disease-free states, which means that
results concerning convergence of terminal values, in for example [1]}, cannot be applied. This explains why only partial results, derived by considering appropriate bounding processes, are proved concerning discontinuity of the final size of the stochastic model. For the special case of the
SI model, in which there is no recovery from infection so infectious individuals remain so forever, this difficulty disappears. Moreover, for the SI model, the deterministic
limiting process can be analysed to yield an explicit non-linear equation for the final size \(\tau \) , which enables us to prove a necessary and sufficient
condition for there to be a discontinuity at the threshold \(\lambda =\lambda _c\) .
